Hi Karin and welcome
.
I've hunted around the Net and finally came up with a site that mentions
an email address for Pussycat Lodge. Scroll down to New Zealand and it gives you the authorised quarantine stations and their contact details. Pussy Cat Lodge must keep their email address very secret as there was no sign of it on any of the other sites I visited, only the address and phone number.
Sorry, can't answer about the birth certificate issue as I haven't had experience of the EOI forms. I know with the ITA you can send in any delayed information after you've already submitted it so maybe it's the same with the EOI. If you're doing it online try it without the number and see what happens.